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SHEAVES OF MICROFUNCTIONS

Andrea D’Agnolo Giuseppe Zampieri

Abstract. Let A be a closed set of M ∼ = R

n

, whose conormal cones x + γ

x

(A), x ∈ A, have locally empty intersection. We first show in §1 that dist(x, A), x ∈ M \ A is a C

1

function. We then represent the microfunctions of C

A|X

, X ∼ = C

n

, (cf [S]), using cohomology groups of O

X

of degree 1. By the results of §1–3, we are able to prove in §4 that the sections of C

A|X

˛

˛

(TMX⊕γ(A))x0

, x

0

∈ ∂A, satisfy the princi- ple of analytic continuation in the complex integral manifolds of {H(φ

Ci

)}

i=1,...,m

, {φ

i

} being a base for the linear hull of γ

x0

(A) in T

x0

M; in particular we get

Γ

MT

MX

(C

A|X

) ˛

˛

˛

∂A×MT˙MX

= 0. In the proof we deeply use a variant of Bochner’s theorem due to [Kan]. When A is a half space with C

ω

boundary, all above results were already proved by Kataoka in [Kat 1]. Finally for a E

X

-module M we show that Hom

EX

(M, C

A|X

)

p

= 0, p ∈ ˙π

−1

(x

0

), when at least one conormal θ ∈ ˙γ

x0

(A) is non-characteristic for M. We also show that for an open domain Ω such that the set A = M \ Ω verifies the above conditions, we have (C

Ω|X

)

T

MX

= H

0

(C

Ω|X

) (cf [S] for the case of A convex).

Un teorema di propagazione per certi fasci di microfunzioni Sunto. Sia A un insieme chiuso di M ∼ = R

n

, i cui coni conormali x + γ

x

(A), x ∈ A, hanno localmente intersezione vuota. Si prova nel §1 che dist(x, A), x ∈ M \ A

`

e una funzione C

1

. Si rappresentano poi le microfunzioni di C

A|X

, X ∼ = C

n

, (cf [S]), mediante gruppi di coomologia di O

X

in grado 1. Se ne deduce nel §4 un principio di prolungamento analitico per sezioni di C

A|X

˛

˛

(TMX⊕γ(A))x0

, x

0

∈ ∂A, che generalizza i risultati di [Kat 1]. (Per la dimostrazione si usa essenzialmente un’idea di [Kan].) Se ne d` a infine applicazione ai problemi ai limiti.

§1. Let X be a C -manifold, A a closed set of X. We denote by γ(A) ⊂ T X the set

γ x (A) = C(A, {x}), x ∈ A,

where C(A, {x}) is the normal cone to A along {x} in the sense of [K-S]; we also denote by γ (A) the polar cone to γ(A). We assume that in some coordinates in a neighborhood of a point x 0 ∈ ∂A:

(i) (x − γ x (A)) ∩ (y − γ y (A)) ∩ S = ∅ ∀x 6= y ∈ ∂A ∩ S, (ii) x 7→ γ x (A) is upper semicontinuous.

(1.1)

Appeared in: Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. (9) Mat. Appl. 1 (1990), no. 1, 53–58

1991 Mathematics Subject Classification. 58 G 20.

Key words and phrases. Partial differential equations on manifolds; boundary value problems..

1

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Remark 1.1.

(a) If A is convex in X ∼ = R n then (1.1) holds. Moreover in this case γ(A) = N (A) where N (A) is the normal cone to A in the linear hull of A.

(b) All sets A with C 2 -boundary satisfy (1.1).

(c) The set A = {(x 1 , x 2 ) ∈ R 2 ; x 1 ≤ −p|x 2 |} verifies (1.1). (Here γ 0 (A) = R x

2

but N 0 (A) = R 2 .)

(d) The set A = {(x 1 , x 2 ) ∈ R 2 ; x 1 ≤ |x 2 | 3/2 } does not verify (1.1); in particular (1.1) is not C 1 -coordinate invariant.

Lemma 1.2. Fix coordinates in X at x 0 and assume that (1.1) holds. Then for every x ∈ (X \ A) ∩ S ε (S ε = {y; |y − x 0 | < ε}, ε small) there exists an unique point a = a(x) ∈ ∂A ∩ S 2ε such that

(1.2) x ∈ a − γ a (A).

Proof. One takes a point a = a(x) verifying

(1.3) |x − a| = dist(x, ∂A),

and verifies easily that a also verifies (1.2). The uniqueness is assured by (1.1).  From the uniqueness it easily follows that a(x) is a continuous function. (One should even easily prove that it is Lipschitz-continuous.)

We set d(x) = dist(x, A) and, for t ≥ 0, A t = {x; d(x) ≤ t}; we also set δ x = γ x (A d(x) ).

Lemma 1.3. Let (1.1) hold in some coordinate system; then δ x , x ∈ ∂A are half- spaces and the mapping x 7→ δ x is continuous.

Proof. We shall show that

(1.4) δ x = {y; hy − x, a(x) − xi ≥ 0}.

(The function x 7→ a(x) being continuous, the lemma will follow at once.) In fact since

{y; |y − a(x)| ≤ d(x)} ⊂ A d(x) ,

then “ ⊇” holds in (1.4). On the other hand we reason by absurd and find a sequence {x ν } such that

(1.5)

 

 

x ν → x, d(x ν ) ≤ d(x),

ha(x) − x, x ν − xi ≤ −δ|x ν − x|, δ > 0.

By continuity we can replace a(x) − x by a(x ν ) − x ν in (1.5) and conclude that, for large ν,

|x − a(x ν ) | < |x ν − a(x ν ) | = d(x ν ) ≤ d(x), a contradiction. 

Let N (A) be the normal cone to A in the sense of [K-S].

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Lemma 1.4. Let B be closed and assume that:

(1.6) γ x (B) is a half space for every x ∈ ∂B, (1.7) x 7→ γ x (B) is continuous.

Then N x (B), x ∈ ∂B are also half spaces.

Proof. Suppose by absurd that there exist Γ 0 ⊂⊂ γ x

0

(B) and two sequences {z ν }, {y ν } with

 

 

 

 

z ν , y ν → x 0 , z ν , y ν ∈ ∂B,

θ ν = y ν − z ν ∈ intΓ 0 , [z ν , y ν ] ⊂ B

(Here [z ν , y ν ] denotes the segment from z ν to y ν .) But then γ y

ν

⊃ Γ 0 ∪ {−θ ν }, a contradiction. 

Remark 1.5. Let B verify N x

0

(B) 6= {0}; then if one takes coordinates with (0, . . . , 0, 1) ∈ N x

0

(B) and sets x = (x 0 , x n ), one can represent ∂B = {x; x n − ϕ(x 0 ) = 0 } for a Lipschitz-continuous function ϕ. Moreover if N x

0

(B) is a half- space and if we let R + (0, . . . , 0, 1) = N x

0

(B) then ϕ is differentiable at x 0 due to

|ϕ(x 0 ) − ϕ(x 0 0 ) | = o(|x 0 − x 0 0 |).

Proposition 1.6. Let (1.1) hold in some coordinates; then d(x), x / ∈ A is a C 1 function.

Proof. By Lemmas 1.3, 1.4, N x (A d(x) ) are half-spaces; set

τ x = ∂N x (A d(x) ) and denote by n x the normal to τ x . Let y ∈ τ x ; according to Remark 1.5 there exists ˜ y ∈ ∂A d(x) with |˜y − y| = o(|y − x|). It follows:

(1.8) |d(y) − d(x)| = |d(y) − d(˜y)| ≤ k|y − ˜y| = o(|y − x|).

By (1.8) we obtain ∂/∂τ x d(x) = 0. On the other hand one has

∂/∂n x d(x) = 1. Finally ∂d(x) = n x , x / ∈ A, and hence d is C 1 . 

§2. Let X be a C -manifold, Y ⊂ X a C 1 -submanifold, M · a complex of Z- modules of finite rank, and set M ·∗ = R Hom Z (M · , Z). Let µhom( ·, ·) be the bifunctor of [K-S, §5]; one easily proves that

(2.1) µhom(Z Y , M Y · ) ∼ = M T ·

Y

X ,

(2.2) µhom(M Y · , Z Y ) ∼ = (M T ·

Y

X ) .

Lemma 2.1. Let M Y · ∼ = Z Y in D + (X; p), p ∈ ˙T Y X ( [K-S, §6]). Then M · ∼ = Z.

Proof. The proof is a straightforward consequence of the formula Hom D

+

(X;p) ( ·, ·) → H 0 µhom( ·, ·) p ,

and of (2.1), (2.2). 

(4)

§3. Let M be a C ω -manifold of dimension n, X a complexification of M , A ⊂ M a closed set. According to [S] we define

C A|X = µhom(Z A , O X ) ⊗ or M |X [n].

We assume here that

(i) A satisfies (1.1) in some coordinates at x 0 = 0, (ii) A = intA in the linear hull of A,

(iii) SS(Z A ) ⊂ γ (A).

(3.1)

We take coordinates (x 0 , x n ) ∈ R n−1 × R ∼ = M , (z 0 , z n ) ∈ C n ∼ = X, and suppose that A = A 0 × R. We define

(3.2) G A = {z; y n ≥ inf

a

0

∈A

0

a 02 /4 + hy 0 , a 0 i/2}.

Lemma 3.1. ∂G A is C 1 . Proof. One defines the set

{z;y n ≥ −y 02 /4 for y 0 ∈ −A 0 ,

y n ≥ a 2 ( −y 0 )/4 + hy 0 , a( −y 0 ) i/2 for y 0 ∈ −A / 0 }, (3.3)

(where a( −y 0 ) is the point of ∂A 0 such that −y 0 ∈ a(−y 0 ) − γ a(−y

0

) (A 0 )).

One easily proves that the above set coincides with G A . Then one observes that the boundary of (3.3) is defined by

(3.4) y n =

 

 

 

 

−y 0 /4, for y 0 ∈ −A 0

a 2 ( −y 0 )/4 + hy 0 , a( −y 0 ) i/2 for y 0 ∈ −A / 0

= |y 0 + a( −y 0 ) | 2 /4 − y 02 /4

= dist 2 (y 0 , −A 0 )/4 − y 02 /4, .

Since dist(y 0 , −A 0 ), y 0 ∈ M \ −A 0 , is C 1 due to Proposition 1.4, then the function defined in (3.4) is also C 1 .

Proposition 3.2. (cf [S])

(i) We can find a complex homogeneous symplectic transformation φ such that (3.5) φ(A × M T M X ⊕ γ (A)) = N (G A ).

(ii) φ can be quantized to Φ such that

(3.6) Φ(Z A ) = Z G

A

[n − 1];

(5)

in particular

(3.7) ( C A|X ) p ∼ = H 1 G

A

( O X ) π(φ(p)) , p ∈ ˙π −1 (x 0 ).

Proof. One takes coordinates (z, ζ) ∈ T X, and defines φ for Im ζ n > 0 by:

 

 

 

 

z 0 7→ ζ 0 /ζ n − √

−1 z 0 , z n 7→ hz, ζ/ζ n i/2 − √

−1 z 02 /4, ζ 0 7→ −z 0 ζ n /2,

ζ n 7→ ζ n .

Then by recalling that G A coincides with the set (3.3), one gets (i). As for (ii) one sets F = Φ(Z A ), Y = ∂G A , and denotes by j : Y , → X the embedding. One gets SS( F) ⊂ N (G A ) ⊂ π −1 (Y ) at p; hence F ∼ = Rj ∗ G at p for some G in D + (Y ) (cf [K-S, §6]). On the other hand one has SS(G) ⊂ T X X at π(p) ([K-S, Prop. 4.1.1]);

hence G ∼ = M Y · at π(p) for a complex of Z-modules.

Due to (3.1)(ii) there exists q ∼ p such that A is a manifold at π(q) and hence by [K-S, §11] we get F ∼ = Z Y [n − 1] at φ(q). Thus (3.6) follows from Lemma 2.1, and (3.7) from the fact that X \ G A is pseudoconvex. 

For convex A, the above proposition is stated in [S].

§4. Let M be a C ω -manifold, X a complexification of M , A ⊂ M a closed set satisfying conditions (3.1).

Proposition 4.1. Let {φ i } i=1,...,m be a base for the space spanned by γ x

0

(A) in T x

0

X. Then the sections of C A|X

(T

M

X⊕γ

(A))

x0

satisfy the principle of the analytic continuation on the complex integral manifolds of {H(φ C i ) } i=1,...,m .

Proof. Using the trick of the dummy variable we can assume A being of the form A 0 × R and hence use the transformation φ of §3. Let p, q ∈ πφ((T M X ⊕ γ (A)) x

0

) belong to the same integral leaf of {H(φ C i ) } i=1,...,m . We then have to show that if f is holomorphic in X \ G A and extends holomorphically at p , then it also extends at q.

We observe that π((T M X ⊕ γ (A)) x

0

) = π(∂G A ∩ φ( ˙π −1 (x 0 )) is plane. Thus the claim follows from the Bochner’s tube theorem at least when ρ M (p) belongs to the interior of γ x

0

(A) in the plane of {φ i } (ρ M : T X → T M ).

Otherwise one has to remember that ∂G A is C 1 , and use the following result whose proof is straightforward.

Lemma 4.2. (cf [Kan]) Let (z 1 , z 2 ) ∈ C 2 , z i = x i + √

−1 y i , i = 1, 2, and let ψ be a C 1 function on R 2 y

1

,y

2

at 0 such that ψ ≥ 0 and ψ = 0 for y 1 ≥ 0. If f is analytic in the set

{|x i | < ε} × ({|y i | < ε, y 2 > ψ(y 1 ) } ∪ {y 1 = ε, −δ < y 2 ≤ 0}),

then f is analytic at 0.

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Remark 4.3.

(a) When A is a half-space with C ω -boundary, Proposition 4.1 was already stated in [Kat 1].

(b) In the situation of Proposition 4.1, one has (cf [Kat 1]):

Γ A×

M

T

M

X ( C A|X )

∂A×

M

T ˙

M

X = 0.

(c) Let M be an E X -module at p ∈ ˙π −1 (x 0 ). Suppose that there exists θ ∈

˙γ x

0

(A) non-characteristic for M. Then:

Hom E

X

( M, C A|X ) p = 0.

(This was announced by Uchida when A is convex and all θ ∈ ˙ N x

0

(A) (or

∂N x

0

(A)) are non-characteristic.)

Let now Ω be an open set of M and assume that A = M \ Ω satisfies the hypotheses (3.1). By the distinguished triangle

C A|X → C M |X → C Ω|X +1

→,

by (3.7), and by the corresponding formula for C M |X , one gets (cf [S]):

Proposition 4.4.

H 0 ( C Ω|X ) = ( C Ω|X ) T

M

X .

By (4.1) and by Remark 4.3 (c), one also gets, for a D X -module M:

Hom D

X

( M, A ) x

0

= {f ∈ Hom D

X

( M, Γ Ω ( B M )) x

0

; SS M,0 (f ) ∩ ˙π −1 (x 0 ) = ∅},

SS M,0 (f ) being the micro-support in the sense of [S]. (One needs perhaps to assume here Z cohomologically constructible; but this follows probably from (1.1).)

References

[Kan] A. Kaneko, Estimation of singular spectrum of boundary values for some semihyperbolic operators, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 27(2) (1980), 401–461.

[K-S] M. Kashiwara, P. Schapira, Microlocal study of sheaves, Asterisque 128 (1985).

[Kat 1] K. Kataoka, On the theory of Radon transformation of hyperfunctions, J. Fac. Sci. Univ.

Tokyo Sect. 1A 28 (1981), 331-413.

[Kat 2] K. Kataoka, Microlocal theory of boundary value problems I, J. Fac. Sci. Univ. Tokyo Sect. 1A 27 (1980), 355-399; II 28 (1981), 31–56.

[S-K-K] M. Sato, M. Kashiwara, T. Kawai, Hyperfunctions and pseudo-differential equations, Lecture Notes in Math., Springer-Verlag 287 (1973), 265–529.

[S] P. Schapira, Front d’onde analytique au bord I, C. R. Acad. Sci. Paris S´ er. I Math. 302 10 (1986), 383–386; II, S´ em. E.D.P. ´ Ecole Polytechnique Exp. 13 (1986).

Dipartimento di matematica, via Belzoni 7, 35131 Padova, Italy

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